Boundaries of Unbounded Fatou Components of Entire Functions

An unbounded Fatou component U of a transcendental entire function is simplyconnected. The paper studies the boundary behaviour of the Riemann map Ψ of the disc D to U , in particular the set Θ of ∂D where the radial limit of Ψ is ∞ . If U is not a Baker domain and ∞ is accessible in U , then Θ is dense in ∂D . If U is a Baker domain in which f is not univalent, Θ contains a non-empty perfect subset of ∂D . Examples show that Θ may be either countably infinite or residual in ∂D . The function f(z) = z + e−z leads to a component U with a particularly interesting prime end structure.